Effects of XY Modulation

Say that your original image does not suffer from wrap around effects, and thus looks as follows:

K-space modulated Image, X-Direction =

The image is simply being shifted to the right, with the right end wrapping around to the left side. If the axis is described as going from 0 to 2pi, the column at 0 is moved to pi, and the column at pi is being moved to 0.

K-space modulated Image, Y-Direction =

In this case the image is simply being shifted towards the top of the screen, with the top of the image wrapping around to the bottom.

K-space modulated Image, X & Y Directions =

Modulating both has the same effect as moving the original image up and to the right, with the image wrapping around from the left and bottom.

Obviously K-space modulation simply equates to shifting the image along the X or Y axis, which allows for a number of imaging conventions. For instance, our raw data produces images that must be Y-space modulated in order to produce a continuous image, while image coming from another brand of MRI might require X-space modulation.

So, why does negating every other element in each row produce shifting in the X direction, and negation of every other element in each column produce shifting in the Y direction?

It is actually a very interesting feature of the Fourier Tranform - reversing the sign of the frequency at every other point in the F.T. is the same thing as multiplying the argument of the F.T. by a sinusoid at the Nyquist frequency!